Coding vertical aysmptote to appear with drag points

I am trying to put together a review activity for graphing reciprocal trig functions. I want the students to drag points to the correct places for the asymptotes and the “vertex” of the individual curves. When the points are in the correct places, the graph and asymptotes will appear. I’ve got the trig curves to appear, but am struggling with the vertical asymptotes. I would really appreciate some suggestions.

Consider the sine and cosine for each. For example, cos x=0 will graph all the asymptotes for tan x since that’s where tangent is undefined.

This calculator screen might work. Drag Vertical Asymptote